Pexider Type Functional Equation Research Articles

Minimum functional equation and some Pexider-type functional equation on any group

<abstract><p>We discuss the solution to the minimum functional equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>for a real-valued function $ \eta: G \to \mathbb{R} $ defined on arbitrary group $ G $. In addition, we examine the Pexider-type functional equation</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \chi(x)\eta(y)+\psi(x), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>where $ \eta $, $ \chi $ and $ \psi $ are real mappings acting on arbitrary group $ G $. We also investigate this Pexiderized functional equation that generalizes two functional equations</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>with the restriction that the function $ \eta $ satisfies the Kannappan condition.</p></abstract>

On the Stability of Mixed Additive--Quadratic and Additive--Drygas Functional Equations

In this paper, we have improved some of the results in [C. Choi and B. Lee, Stability of Mixed Additive-Quadratic and Additive--Drygas Functional Equations. Results Math. 75 no. 1 (2020), Paper No. 38]. Indeed, we investigate the Hyers-Ulam stability problem of the following functional equationsbegin{align*} 2varphi(x + y) + varphi(x - y) &= 3varphi(x)+ 3varphi(y) \ 2psi(x + y) + psi(x - y) &= 3psi(x) + 2psi(y) + psi(-y).end{align*}We also consider the Pexider type functional equation [2psi(x + y) + psi(x - y) = f(x) + g(y),] and the additive functional equation[2psi(x + y) + psi(x - y) = 3psi(x) + psi(y).]

Superstability and Solution of The Pexiderized Trigonometric Functional Equation

The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.

SUPERSTABILITY OF A GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATION

We will investigate the superstability of the trigonometric functional equation from the following Pexider type functional equation: f ( x + y ) − f ( x − y ) = λ · g ( x ) h ( y ) , λ is constant, which is a trigonometric functional equation mixed by the sine and cosine function. More- over, the equation can be considered by the mixed functional equation of the hyperbolic trigonometric functions, several exponential type functions, and Jensen type equation.

Alienation of the Quadratic and Additive Functional Equations

Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation (*) $$f(x+y)+f(x-y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y),$$ resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive one (alienation phenomenon). Moreover, some stability result for equation (*) is also presented.

Stability of a Pexider type functional equation related to distance measures

This work aims to study of the stability of two generalizations of the functional equa- tion f(pr,qs)+f(ps,qr )= f(p,q) f(r,s), namely (i) f(pr,qs)+g(ps,qr )= h(p,q)h(r,s) ,a nd (ii) f(pr,qs)+g(ps,qr )= h(p,q)k(r,s) for all p,q,r,s ∈ G ,w hereG is a commutative semi- group. Thus this work is a continuation of our earlier works (15 )a nd (16), and the functional equations studied here arise in the characterizations of symmetrically compositive sum form distance measures.

The superstability of a variant of Wilson’s functional equation on an arbitrary group

The aim of this paper is to investigate the superstability problem for the variant $$\begin{aligned} f(xy)+f(\sigma (y)x)=2f(x)g(y),\ \ \ \ x,y\in G, \end{aligned}$$ of Wilson’s functional equation where \(G\) is an arbitrary group, \(f,g\) are complex valued functions and \(\sigma \) is an involution of \(G\). As a consequence, we obtain the superstability of the Pexider type functional equation $$\begin{aligned} f(xy)+f(\sigma (y)x)=2g(x)h(y),\ \ x,y\in G, \end{aligned}$$ on any group.

On the Superstability of Trigonometric Type Functional Equations

The aim of this paper is to study the superstability for the mixed trigonometric functional equation: , , ), ( ) ( 2 )) ( ( ) ( G y x y g x f y x f xy f ∈ = − σ ) ( ,g f E and the stability of the Pexider type functional equation: , , ), ( ) ( 2 )) ( ( ) ( G y x y h x g y x f xy f ∈ = − σ ) ( , , h g f E where G is any group, not necessarily abelian, g f , and h are unknown complex valued functions and σ is an involution of G . As a consequence we prove that if f satisfies the inequality δ σ ≤ − − ) ( ) ( 2 )) ( ( ) ( y f x f y x f xy f for all G y x ∈ , then f is bounded.

ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART I

AbstractWe investigate the Pexider-type functional equation where f,g,h are real functions defined on an abelian group G.

ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

Additivity and exponentiality are alien to each other

Let (S, +) be a (semi)group and let (R,+, ·) be an integral domain. We study the solutions of a Pexider type functional equation $$f(x+y) + g(x+y) = f(x) + f(y) + g(x)g(y)$$ for functions f and g mapping S into R. Our chief concern is to examine whether or not this functional equation is equivalent to the system of two Cauchy equations $$\left\{\begin{array}{@{}ll} f(x+y) = f(x) + f(y)\\ g(x+y) = g(x)g(y)\end{array}\right.$$ for every \({x,y \in S}\).

Superstability of Some Pexider-Type Functional Equation

We will investigate the superstability of the sine functional equation from the following Pexider-type functional equation (), which can be considered the mixed functional equation of the sine and cosine functions, the mixed functional equation of the hyperbolic sine and hyperbolic cosine functions, and the exponential-type functional equations.

On the Stability of a New Pexider-Type Functional Equation

We establish the generalized Hyers-Ulam stability of a Pexider-type functonal equation , which is mixed of a quadratic and an additive functional equations. Also, we obtain its general solution from the stability results.

The stability of a Wilson type and a Pexider type functional equation

The stability of a Wilson type and a Pexider type functional equation

On a pexider-type functional equation for quasideviation means

On a pexider-type functional equation for quasideviation means